Question: $ D = \left[\begin{array}{rrr}-1 & 0 & -1 \\ 0 & 3 & 3\end{array}\right]$ $ C = \left[\begin{array}{rr}-2 & 0 \\ 0 & 2 \\ 1 & 5\end{array}\right]$ What is $ D C$ ?
Answer: Because $ D$ has dimensions $(2\times3)$ and $ C$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ D C = \left[\begin{array}{rrr}{-1} & {0} & {-1} \\ {0} & {3} & {3}\end{array}\right] \left[\begin{array}{rr}{-2} & \color{#DF0030}{0} \\ {0} & \color{#DF0030}{2} \\ {1} & \color{#DF0030}{5}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ D$ , with the corresponding elements in column $j$ of the second matrix, $ C$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ D$ with the first element in ${\text{column }1}$ of $ C$ , then multiply the second element in ${\text{row }1}$ of $ D$ with the second element in ${\text{column }1}$ of $ C$ , and so on. Add the products together. $ \left[\begin{array}{rr}{-1}\cdot{-2}+{0}\cdot{0}+{-1}\cdot{1} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ D$ with the corresponding elements in ${\text{column }1}$ of $ C$ and add the products together. $ \left[\begin{array}{rr}{-1}\cdot{-2}+{0}\cdot{0}+{-1}\cdot{1} & ? \\ {0}\cdot{-2}+{3}\cdot{0}+{3}\cdot{1} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ D$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ C$ and add the products together. $ \left[\begin{array}{rr}{-1}\cdot{-2}+{0}\cdot{0}+{-1}\cdot{1} & {-1}\cdot\color{#DF0030}{0}+{0}\cdot\color{#DF0030}{2}+{-1}\cdot\color{#DF0030}{5} \\ {0}\cdot{-2}+{3}\cdot{0}+{3}\cdot{1} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{-1}\cdot{-2}+{0}\cdot{0}+{-1}\cdot{1} & {-1}\cdot\color{#DF0030}{0}+{0}\cdot\color{#DF0030}{2}+{-1}\cdot\color{#DF0030}{5} \\ {0}\cdot{-2}+{3}\cdot{0}+{3}\cdot{1} & {0}\cdot\color{#DF0030}{0}+{3}\cdot\color{#DF0030}{2}+{3}\cdot\color{#DF0030}{5}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}1 & -5 \\ 3 & 21\end{array}\right] $